Questions - SPS - A-Level Maths

SPS SPS FM Pure 2022 June Q3

3. (a) Show on an Argand diagram the locus of points given by $$| z - 10 - 12 i | = 8$$ Set $A$ is defined by $$A = \left\{ z : 0 \leqslant \arg ( z - 10 - 10 i ) \leqslant \frac { \pi } { 2 } \right\} \cap \{ z : | z - 10 - 12 i | \leqslant 8 \}$$ (b) Shade the region defined by $A$ on your Argand diagram.
(c) Determine the area of the region defined by $A$.
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SPS SPS FM Pure 2022 June Q4

4. The curve with equation $y = \mathrm { f } ( x )$ where $$f ( x ) = x ^ { 2 } + \ln \left( 2 x ^ { 2 } - 4 x + 5 \right)$$ has a single turning point at $x = \alpha$

Show that $\alpha$ is a solution of the equation $$2 x ^ { 3 } - 4 x ^ { 2 } + 7 x - 2 = 0$$ The iterative formula $$x _ { n + 1 } = \frac { 1 } { 7 } \left( 2 + 4 x _ { n } ^ { 2 } - 2 x _ { n } ^ { 3 } \right)$$ is used to find an approximate value for $\alpha$.
Starting with $x _ { 1 } = 0.3$
calculate, giving each answer to 4 decimal places,
1. the value of $x _ { 2 }$
2. the value of $x _ { 4 }$ Using a suitable interval and a suitable function that should be stated,
show that $\alpha$ is 0.341 to 3 decimal places.
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SPS SPS FM Pure 2022 June Q5

5. The triangle $T$ has vertices at the points $( 1 , k ) , ( 3,0 )$ and $( 11,0 )$, where $k$ is a positive constant. Triangle $T$ is transformed onto the triangle $T ^ { \prime }$ by the matrix $$\left( \begin{array} { c c } 6 & - 2
1 & 2 \end{array} \right)$$ Given that the area of triangle $T ^ { \prime }$ is 364 square units, find the value of $k$.
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SPS SPS FM Pure 2022 June Q6

6. The complex number $w$ is given by $$w = 10 - 5 \mathrm { i }$$

Find $| w |$.
Find arg $w$, giving your answer in radians to 2 decimal places. The complex numbers $z$ and $w$ satisfy the equation $$( 2 + \mathrm { i } ) ( z + 3 \mathrm { i } ) = w$$
Use algebra to find $z$, giving your answer in the form $a + b \mathrm { i }$, where $a$ and $b$ are real numbers. Given that $$\arg ( \lambda + 9 i + w ) = \frac { \pi } { 4 }$$ where $\lambda$ is a real constant,
find the value of $\lambda$.
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SPS SPS FM Pure 2022 June Q7

7. \section*{Figure 1}

\includegraphics[max width=\textwidth, alt={}]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-16_634_1025_191_479}

Figure 1 shows the finite region $R$, which is bounded by the curve $y = x \mathrm { e } ^ { x }$, the line $x = 1$, the line $x = 3$ and the $x$-axis. The region $R$ is rotated through 360 degrees about the $x$-axis.
Use integration by parts to find an exact value for the volume of the solid generated.
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SPS SPS FM Pure 2022 June Q8

8. With respect to a fixed origin $O$, the line $l$ has equation $$\mathbf { r } = \left( \begin{array} { c } 13
8
1 \end{array} \right) + \lambda \left( \begin{array} { r } 2
2
- 1 \end{array} \right) \text {, where } \lambda \text { is a scalar parameter. }$$ The point $A$ lies on $l$ and has coordinates $( 3 , - 2,6 )$.
The point $P$ has position vector ( $- \boldsymbol { i } + 2 \boldsymbol { k }$ ) relative to $O$.
Given that vector $\overrightarrow { P A }$ is perpendicular to $l$, and that point $B$ is a point on $l$ such that $\angle B P A = 45 ^ { \circ }$, find the coordinates of the two possible positions of $B$.
(5)
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SPS SPS FM Pure 2022 June Q9

9. Prove by induction that for $n \in \mathbb { Z } ^ { + }$ $$\mathrm { f } ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21
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SPS SPS FM Pure 2022 June Q10

10. The curve defined by the parametric equations $$x = 2 \cos \theta , y = 3 \sin ( 2 \theta ) \text { and } \theta \in [ 0,2 \pi ]$$ is shown below.
The point $P \left( \sqrt { 3 } , \frac { 3 \sqrt { 3 } } { 2 } \right)$ is marked on the curve.
\includegraphics[max width=\textwidth, alt={}, center]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-22_604_826_518_758}

Show that the equation of the normal to the curve at $P$ can be written as $3 y - x = \frac { 7 \sqrt { 3 } } { 2 }$
Show that the Cartesian equation of the curve may be written as $a y ^ { 2 } + b x ^ { 4 } + c x ^ { 2 } = 0$ where $a$, $b$ and $c$ are integers to be found.
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SPS SPS FM Pure 2022 June Q11

11. Solve the differential equation $$2 \cot x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 4 - y ^ { 2 } \right)$$ for which $y = 0$ at $x = \frac { \pi } { 3 }$, giving your answer in the form $\sec ^ { 2 } x = \mathrm { g } ( y )$.
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SPS SPS FM Pure 2022 June Q13

13
8
1 \end{array} \right) + \lambda \left( \begin{array} { r } 2
2
- 1 \end{array} \right) \text {, where } \lambda \text { is a scalar parameter. }$$ The point $A$ lies on $l$ and has coordinates $( 3 , - 2,6 )$.
The point $P$ has position vector ( $- \boldsymbol { i } + 2 \boldsymbol { k }$ ) relative to $O$.
Given that vector $\overrightarrow { P A }$ is perpendicular to $l$, and that point $B$ is a point on $l$ such that $\angle B P A = 45 ^ { \circ }$, find the coordinates of the two possible positions of $B$.
(5)
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9. Prove by induction that for $n \in \mathbb { Z } ^ { + }$ $$\mathrm { f } ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21
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10. The curve defined by the parametric equations $$x = 2 \cos \theta , y = 3 \sin ( 2 \theta ) \text { and } \theta \in [ 0,2 \pi ]$$ is shown below.
The point $P \left( \sqrt { 3 } , \frac { 3 \sqrt { 3 } } { 2 } \right)$ is marked on the curve.
\includegraphics[max width=\textwidth, alt={}, center]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-22_604_826_518_758}

Show that the equation of the normal to the curve at $P$ can be written as $3 y - x = \frac { 7 \sqrt { 3 } } { 2 }$
Show that the Cartesian equation of the curve may be written as $a y ^ { 2 } + b x ^ { 4 } + c x ^ { 2 } = 0$ where $a$, $b$ and $c$ are integers to be found.
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11. Solve the differential equation $$2 \cot x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 4 - y ^ { 2 } \right)$$ for which $y = 0$ at $x = \frac { \pi } { 3 }$, giving your answer in the form $\sec ^ { 2 } x = \mathrm { g } ( y )$.
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12. A linear transformation T of the $x - y$ plane has an associated matrix $\mathbf { M }$, where $\mathbf { M } = \left( \begin{array} { l c } \lambda & k
1 & \lambda - k \end{array} \right)$, and $\lambda$
and $k$ are real constants. and $k$ are real constants.
You are given that $\operatorname { det } \mathbf { M } > 0$ for all values of $\lambda$.
1. Find the range of possible values of $k$.
2. What is the significance of the condition $\operatorname { det } \mathbf { M } > 0$ for the transformation T ? For the remainder of this question, take $k = - 2$.
Determine whether there are any lines through the origin that are invariant lines for the transformation T .
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13. (i) Show that $\sin \left( 2 \theta + \frac { 1 } { 2 } \pi \right) = \cos 2 \theta$.
(ii) Hence solve the equation $\sin 3 \theta = \cos 2 \theta$ for $0 \leqslant \theta \leqslant 2 \pi$.
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SPS SPS FM Pure 2022 June Q14

14. Using an appropriate substitution, or otherwise, show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin 2 \theta } { 1 + \cos \theta } d \theta = 2 - 2 \ln 2$$ [BLANK PAGE]
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SPS SPS SM Pure 2022 June Q1

6 marks

1.

The expression $\left( 2 + x ^ { 2 } \right) ^ { 3 }$ can be written in the form $$8 + p x ^ { 2 } + q x ^ { 4 } + x ^ { 6 }$$ Demonstrate clearly, using the binomial expansion, that $p = 12$ and find the value of $q$.
[0pt] [3 marks]
Hence find $\int \frac { \left( 2 + x ^ { 2 } \right) ^ { 3 } } { x ^ { 4 } } \mathrm {~d} x$.
[0pt] [3 marks]
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SPS SPS SM Pure 2022 June Q2

2. The trapezium $A B C D$ is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{8053bd07-c2b2-4ada-ae0e-8ab6b8466c78-06_296_586_237_778} The line $A B$ has equation $2 x + 3 y = 14$ and $D C$ is parallel to $A B$. The point D has coordinates ( 3,7 ).

Find an equation of the line DC
(2 marks)
The angle $B A D$ is a right angle. Find an equation of the line $A D$, giving your answer in the form $m x + n y + p = 0$, where $m , n$ and $p$ are integers.
(3 marks)
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SPS SPS SM Pure 2022 June Q3

3. A circle has centre $C ( 3 , - 8 )$ and radius 10.

Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
Find the $x$-coordinates of the points where the circle crosses the $x$-axis.
The line with equation $y = 2 x + 1$ intersects the circle.
1. Show that the $x$-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + 6 x - 2 = 0$$
2. Hence show that the $x$-coordinates of the points of intersection are of the form $m \pm \sqrt { n }$, where $m$ and $n$ are integers.
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SPS SPS SM Pure 2022 June Q4

4. The function f is defined by $$f ( x ) = \frac { 5 x } { 7 x - 5 }$$

The domain of f is the set $\{ x \in \mathbb { R } : x \neq a \}$ State the value of $a$
Prove that f is a self-inverse function
Find the range of f
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SPS SPS SM Pure 2022 June Q5

5. Relative to a fixed origin $O$,

the point $A$ has position vector $- 2 \mathbf { i } + 3 \mathbf { j }$,
the point $B$ has position vector $3 \mathbf { i } + p \mathbf { j }$, where $p$ is constant,

Given that $| \overrightarrow { A B } | = 5 \sqrt { 2 }$, find the possible values for $p$.
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SPS SPS SM Pure 2022 June Q6

6. A small company which makes batteries for electric cars has a 10 year plan for growth. In year 1 the company will make 2600 batteries.
In year 10 the company aims to make 12000 batteries.
In order to calculate the number of batteries it will need to make each year from year 2 to year 9, the company considers two models. Model $A$ assumes that the number of batteries it will make each year will increase by the same number each year.

According to model $A$, determine the number of batteries the company will make in year 2 . Give your answer to the nearest whole number of batteries. Model $B$ assumes that the numbers of batteries it will make each year will increase by the same percentage each year.
According to model $B$, determine the number of batteries the company will make in year 2 . Give your answer to the nearest 10 batteries. Sam calculates the total number of batteries made from year 1 to year 10 inclusive, using each of the two models.
Calculate the difference between the two totals, giving your answer to the nearest 100 batteries.
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SPS SPS SM Pure 2022 June Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8053bd07-c2b2-4ada-ae0e-8ab6b8466c78-16_504_951_199_578} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Diagram not drawn to scale Figure 1 shows the plan view of a design for a stage at a concert.
The stage is modelled as a sector $B C D F$, of a circle centre $F$, joined to two congruent triangles $A B F$ and $E D F$. Given that $A F E$ is a straight line, $A F = F E = 10.7 \mathrm {~m} , B F = F D = 9.2 \mathrm {~m}$ and angle $B F D = 1.82$ radians, find the perimeter of the stage, in metres, to one decimal place,
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SPS SPS SM Pure 2022 June Q8

4 marks

8. The function $\mathrm { f } ( x )$ is such that $\mathrm { f } ( x ) = - x ^ { 3 } + 2 x ^ { 2 } + k x - 10$ The graph of $y = \mathrm { f } ( x )$ crosses the $x$-axis at the points with coordinates $( a , 0 ) , ( 2,0 )$ and $( b , 0 )$ where $a < b$

Show that $k = 5$
[0pt] [1 mark]
Find the exact value of $a$ and the exact value of $b$
[0pt] [3 marks]
The functions $\mathrm { g } ( x )$ and $\mathrm { h } ( x )$ are such that $$\begin{aligned} & g ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 10
& h ( x ) = - 8 x ^ { 3 } + 8 x ^ { 2 } + 10 x - 10 \end{aligned}$$
1. Explain how the graph of $y = \mathrm { f } ( x )$ can be transformed into the graph of $y = \mathrm { g } ( x )$ Fully justify your answer.
(ii) Explain how the graph of $y = \mathrm { f } ( x )$ can be transformed into the graph of $y = \mathrm { h } ( x )$ Fully justify your answer.
[0pt] [BLANK PAGE] A geometric series has second term 16 and fourth term 8 All the terms of the series are positive. The $n$th term of the series is $u _ { n }$
Find the exact value of $\sum _ { n = 5 } ^ { \infty } u _ { n }$
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SPS SPS SM Pure 2022 June Q10

10.

Sketch the graph of $y = \cos x$ in the interval $0 \leqslant x \leqslant 2 \pi$. State the values of the intercepts with the coordinate axes.
1. Given that $$\sin ^ { 2 } \theta = \cos \theta ( 2 - \cos \theta )$$ prove that $\cos \theta = \frac { 1 } { 2 }$.
2. Hence solve the equation $$\sin ^ { 2 } 2 x = \cos 2 x ( 2 - \cos 2 x )$$ in the interval $0 \leqslant x \leqslant \pi$
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SPS SPS SM Pure 2022 June Q11

11. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 600
u _ { n + 1 } & = p u _ { n } + q \end{aligned}$$ where $p$ and $q$ are constants.
It is given that $u _ { 2 } = 500$ and $u _ { 4 } = 356$

Find the two possible values of $u _ { 3 }$
When $u _ { n }$ is a decreasing sequence, the limit of $u _ { n }$ as $n$ tends to infinity is $L$. Write down an equation for $L$ and hence find the value of $L$.
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SPS SPS SM Pure 2022 June Q12

5 marks

12. A curve is defined for $x \geq 0$ by the equation $$y = 6 x - 2 x ^ { \frac { 3 } { 2 } }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
[0pt] [2 marks]
The curve has one stationary point. Find the coordinates of the stationary point and determine whether it is a maximum or minimum point. Fully justify your answer.
[0pt] [3 marks]
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SPS SPS SM Pure 2022 June Q13

13. $$\frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \equiv A + \frac { B } { ( x - 3 ) } + \frac { C } { ( 1 - 2 x ) }$$ Find the values of the constants $A , B$ and $C$.
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SPS SPS SM Pure 2022 June Q14

14. A region, R , is defined by $x ^ { 2 } - 8 x + 12 \leq y \leq 12 - 2 x$
a) Sketch a graph to show the region $R$. Shade the region $R$.
b) Find the area of R
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SPS SPS SM Pure 2022 June Q15

15. a) Prove that $$n - 1 \text { is divisible by } 3 \Rightarrow n ^ { 3 } - 1 \text { is divisible by } 9$$ b) Show that if $\log _ { 2 } 3 = \frac { p } { q }$, then $$2 ^ { p } = 3 ^ { q }$$ Use proof by contradiction to prove that $\log _ { 2 } 3$ is irrational.
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